International Journal of Scientific & Engineering Research, Volume 3, Issue 1, January-2012 1

ISSN 2229-5518

Solving Blasius Problem by Adomian

Decomposition Method

V. Adanhounme, F.P. Codo

Abstract - Using the Adomian decomposition method we solved the Blasius problem for boundary-layer flows of pure fluids (non-porous domains) over a flat plate. We obtained the velocity components as sums of convergent series. Furthermore we constructed the interval of admissible values of the shear-stress on the plate surface.

Index Terms - Convergent series, Decomposition technique, Fluid flow, Shear-stress.

——————————  ——————————

Nomenclature

1. u velocity in the x-direction
2. u0
velocity of the free stream
3. v velocity in the y-direction
4. x horizontal coordinate
5. y vertical coordinate

6. viscosity coefficient

7. ρ density

8. kinematic viscosity of the fluid

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ISSN 2229-5518

1 INTRODUCTION

The problem of flow past a flat plate is one of interesting problems in fluid mechanics which was first solved by Blasius [5] by assuming a series solutions . Later, numerical methods were used in [7] to obtain the solution of the boundary layer equation. In [2] the first derivative with respect to y of the velocity component
In this section we provide the analytical solutions ,i.e. the fluid velocity components as sums of convergent series using the Adomian decomposition technique and compute the admissible values of the shear-stress on the plate surface.

Consider the stream function

in the x direction at the point

y = 0

for the Blasius

, (5)

problem is computed numerically for the estimation of the shear-stress on the plate surface. Later in [9] one solved the problem above by assuming a finite power series where the objective is to determine the power series coefficients.
The purpose of this study is to obtain the solutions for the Blasius problem for two dimensional boundary layer using the Adomian decomposition technique and to compute the admissible values of the shear-stress on the wall, imposing the constraint on the first derivative with respect to y of the velocity component in the x direction
Where f is a function three times continuously differentiable on the interval and a constant positive real. Then (1) and (2) with (3) are transformed as

, f , (6)

where ( . )’ stands for

Definition 3.1

The problem (6) is called the Blasius problem for boundary-

at the point

y = 0 .

layer flows of pure fluids (non-porous domains) over a flat plate.

2 MATHEMATICAL MODEL

The physical model considered here consists of a flat plate parallel to the x - axis with its leading edge at x = 0 and infinitely long down stream with constant
Let us transform (6) into the nonlinear integral equation. For this purpose, setting we can write the equation in (6) as

(7)

component

u0 of the velocity. For the mathematical

analysis we assume the properties of the fluid such as viscosity and conductivity, to a first approximation, are constant. Under these assumptions the basic equations required for the analysis of the physical phenomenon are the equations of continuity and motion. According to the Boussinesq approximation these equations get the following expressions [2]

(1)

Multiplying by and integrating the result from to
we reduce (7) to

, where (8)

Integrating three times (8) from to and taking into account the boundary conditions in (6) we reduce (8) to the nonlinear integral equation

(2)

with the boundary conditions imposed on the flow in [2]

, , (3)

Where is a stream function related to the velocity components as:

, (4)

3 ANALYTICAL SOLUTION and CONVERGENCE RESULTS

(9)

which is a functional equation

, where (10)

, (11)

(12)

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ISSN 2229-5518

Here

N (g )is a nonlinear operator from a Hilbert space

H into H . In [4] G. Adomian has developed a decomposition technique for solving nonlinear functional equation such as (10). We assume that (10) has a unique solution. The Adomian technique allows us to find the solution of (10) as an infinite series using the following scheme:

, where

The proofs of convergence of the series and

are given below. Without loss of generality we set a0 = 0 and we have the following scheme:

i.e (21)

where the bn are real numbers. Then we obtain

We arrive at the following result

Lemma 4.1

The admissible values of the shear-stress

(18) (19)

(13) (14) (15)

(16) (17)
on the plate surface belong to the open interval

(22)

for each given value of and for the given approximation precision depending on

5 CONCLUSION

In this paper, we have investigated the analytical solutions for the Blasius problem which are the sums of convergent series, using the Adomian decomposition technique. Then we estimated the error by approximating the exact values of the shear-stress on the plate surface obtained in this paper by the approximate values of the shear-stress obtained in [2]. Doing so, we constructed the interval of admissible values of the shear-stress on the plate surface.

By induction, we have

. (20)

REFERENCES

[1] F.Busse,"On the optimum theory of turbulence, Energy Stability and convection", Pitman Research Notes in Mathematics, edited by G. Galdi and B. Straughan, Wiley, New York,1987.

[2]M.E.Eglit et al,"Problemes de mecanique des milieux continus".Tomes 1,2, Lycee Moscovite,1996.

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[3] G.Adomian,"A review of the decomposition method and some recent results for nonlinear equations". Comput.Math.Applic,21(5)101-

127,1991.

[4]G.Adomian,"A review of the decomposition method and some recent results for nonlinear equations", Math.Comp.Modelling

13(7),17-43,1990.

[5]H.Blasius,"Grenzschichten inFlussigkeiten mit kleiner Reibung".

Z.Math.Phys.,56(1),1908.

[6]Y.Cherruault,"Convergence of Adomian's method",

Kybernetes,18(2),31-38,1989.

[7]L.Howarth,"On the solution of the laminar boundary layer equations". Proc.Roy.Soc.London,A164,547,1938.

[8]L.Lu, C.R.Doering,and F.H.Busse,"Bounds on convection driven by internal heating". J. Math.Phys.,45,2967-2986,2004.

[9]P.Vadasz,"Free convection in rotating porous media.Transport

Phenomena in porous media". Pergamon Elsevier Science, 285-

312,1997.

[10]E.Zadrzynska and W.M.Zajaczkowski,"Global Regular Solutions with Large Swirl to the Navier-Stokes Equations in a cylinder". J.M ath.Fluid Mechanics,Vol.11,126-169,2009.

 Francois de Paule Codo received the Mining Engineer, M.Sc. and Ph.D.degrees from the Heavy Industries Technical University of Miskolc, Hungary

He is currently Assistant Professor of Applied Fluid Mechanics and Hydraulics in Department of Civil Engineering at the University of Abomey-Calavi,

Benin.

His principal research interests are applied fluid mechanics and hydraulics at the Applied Mechanics and Energy Laboratory.

e-mail:fdepaule2003@yahoo.fr

 Villevo Adanhounme received the M.Sc. and Ph.D. degrees in Mathematics from the Russian People University of Moscow, Federation of Russia.

He is currently Assistant Professor of Variational Calculus and Advanced Probability at the International Chair of Mathematical Physics and Applications-University of Abomey-Calavi, Benin.

His principal research interests are applied mechanics, partial differential equations and optimal control in the International Chair of Mathematical Physics and Applications.